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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Random multiplicative processes</h3>

<p class="header_title">Introduction</p>

<p> Examples of
random multiplicative processes include the distributions of incomes, rainfall, and
fragment sizes in rock crushing processes. Consider the latter for
which we begin with a rock of size w. We strike the rock with a
hammer and generate two fragments whose sizes are pw and qw,
where q = 1 - p. In the next step the possible sizes of the
fragments are p<sup>2</sup>w, pqw, qpw, and
q<sup>2</sup>w. What is the distribution of the fragments after N blows
of the hammer?</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;To answer this question, consider a binary sequence in which the
numbers
x<sub>1</sub> and
x<sub>2</sub> appear independently with probabilities p and q
respectively. If there are N elements in the product &#928;, we
can ask what is &lt;&#928;&gt;, the mean value of
&#928;? To compute &lt;&#928;&gt;, we define
P(n) as the probability that the product of N independent
factors of x<sub>1</sub> and x<sub>2</sub> has the value x<sub>1</sub><sup>n</sup> x<sub>2</sub><sup>N-n</sup>.
This probability is given by the number of sequences where x<sub>1</sub>
appears
n times multiplied by the probability of choosing a specific
sequence with
x<sub>1</sub> appearing n times:</p>
<p class="center">
P(n) = N!/(n! (N - n)!) p<sup>n</sup> q<sup>N-n</sup>.
</p>
<p>The mean value of the product is given
by</p>
<p class="center">
&lt;&#928;&gt;<sub>mp</sub> = &#931;<sub>n=0</sub> P(n) x<sub>1</sub><sup>n</sup>x<sub>2</sub><sup>N-n</sup>
= (px<sub>1</sub> + qx<sub>2</sub>)<sup>n</sup>.
</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The most probable event is one in which the
product contains Np factors of x<sub>1</sub> and Nq factors of x<sub>2</sub>.
Hence, the most probable value of the product is</p>
<p class="center">
&#928; = (x<sub>1</sub><sup>p</sup>x<sub>2</sub><sup>q</sup>)<sup>N</sup>.
</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.productprocess.ProductProcessApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Problems</p>

<ol>

<li>The average value of the sum of random variables
is a good approximation to the most probable value of the sum.
Is there is a similar relation for a random multiplicative
process? First consider x<sub>1</sub> = 2, x<sub>2</sub> = 1/2, and
p = q = 1/2. Determine &lt;&#928;&gt; and &lt;&#928;&gt;<sub>mp</sub> s.</li>

<li>Use the program to estimate &lt;&#928;&gt; and &lt;&#928;&gt;<sub>mp</sub> for the same parameters as used to calculate the results analytically. Do your estimated values converge more or less
uniformly to the exact values as the number of measurements
becomes large? Do a similar simulation for N = 20. Compare your
results with a similar simulation of a random walk and discuss the
importance of extreme events for random multiplicative processes.</li>

</ol>

<p class="header_title">References</p>

<ul>

<li>S. Redner, <i>Random multiplicative processes: An elementary tutorial,</i> 
Am. J. Phys. <b>58</b>, 267&#8211;273 (1990).</li>

</ul>

<p class="header_title">Java Classes</p>
<ul>

<li>ProductProcessApp</li>

</ul>

<p class = "small">Updated 21 February 2007.</p>
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